We present detailed clustering measurements from the 2dF QSO Redshift Survey (2QZ) in the redshift range 0.8<z<2.1. Using a flux limited sample of ~14,000 objects with effective redshift z_eff=1.47, we estimate the quasar projected correlation function for separations 1<r/h^{-1} Mpc<20. We find that the 2-point correlation function in real space is well approximated by a power law with slope gamma=1.5 pm 0.2 and comoving correlation length r_0=4.8^{+0.9}_{-1.5} h^{-1} Mpc. Splitting the sample into three subsets based on redshift, we find evidence for an increase of the clustering amplitude with lookback time. For a fixed gamma, evolution of r_0 is detected at the 3.6 sigma confidence level. The ratio between the quasar correlation function and the mass autocorrelation function (derived adopting the concordance cosmological model) is found to be scale independent. For a linear mass-clustering amplitude sigma_8=0.8, the ``bias parameter'' decreases from b~3.9$ at z_eff=1.89 to b~1.8 at z_eff=1.06. From the observed abundance and clustering, we infer how quasars populate dark-matter haloes of different masses. We find that 2QZ quasars sit in haloes with M>10^{12} M_sun and that the mean mass of their host haloes is of the order of 10^{13} M_sun. The observed clustering is consistent with assuming that the locally observed correlation between black-hole mass and host-galaxy circular velocity is still valid at z>1. From the fraction of haloes which contain active quasars, we infer that the characteristic quasar lifetime is t_Q ~ a few x 10^7 yr at z~1 and approaches 10^8 yr at higher redshifts. Comment: 23 pages,11 figures, accepted for publication in MNRAS